For a given number field \(K\) and a positive integer \(g\geq 2\) there exists a number \(N(K,g) \) depending only on \(K\) and \(g\) such that for any algebraic curve \(C\) defined over \(K\) having genus equal to \(g\) has at most \(N(K,g) \) \(K\)-rational points.
Caporaso, Harris, and Mazur [1] showed that the conjecture is true under the assumption of the Bombieri-Land Conjecture.
Q808001 Bombieri-Land Conjecture
Q103002 Erdős–Ulam Problem
Q809002 Mazur's Conjecture B
[1] Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society. 10 (1): 1–35. doi:10.1090/S0894-0347-97-00195-1.
[2] Stoll, Michael (2019). "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank". Journal of the European Mathematical Society. 21 (3): 923–956. doi:10.4171/JEMS/857.
[3] Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David (2016). "Uniform bounds for the number of rational points on curves of small Mordell–Weil rank". Duke Mathematical Journal. 165 (16): 3189–3240. arXiv:1504.00694. doi:10.1215/00127094-3673558.
uniform boundedness; genus; algebraic curve; number field